Announcements°HW 5 due Thursday 11/21°Plan ahead: HW 6 will be assigned right before Thanksgivingbreak, due Thursday, Dec. 5°I will cover HW 6 during problem session next week°And also I will bring snacks°Extra credit auctions ongoing°Final Wednesday, Dec. 11, 12:15-3:15pm in NVIDIAAuditorium°Let us know if you have a con±ict or require an examaccomodation2 / 45

Nonlinear Duality and the Lagrange FunctionThen the dual of the general optimization problem is:maximizeφ(λ,y)subject toλ≥0The necessary conditions for optimality of the general primal anddual are just the KKT conditions.Recall that if the primal problem is additionally a convexoptimization problem, then the KKT conditions are also suﬃcientfor optimality.5 / 45

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Nonlinear Dual ExampleSo we get that the dual of:minimizex1+2x2s.t.x21+x22≥1is:maximizeλ+5/(4λ)s.t.λ≥08 / 45

Bisection MethodWe want to fnd the minimum oF a continuously di±erentiableFunctionf:R→Rgiven that the minimizerx∗Falls in someinterval [a,b]suchthatf°(a)f°(b)<0°Looking For point wheref°(x)=0°Search by repeatedly bisecting the search interval°At each step, check midpoint and keep halF oF interval whichcontainsx∗°End when either:°we fnd anx∗such thatf°(x∗) = 0, or°the length oF the search interval|x−x|<°9 / 45

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